1988 Fiscal Year Final Research Report Summary
The investigations on the geometric structures on differentiable manifolds.
| Project/Area Number |
62540040
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Kyoto University |
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Principal Investigator |
TANDAI Kouichi Kyoto University Yoshida College, Professor, 教養部, 教授 (90026732)
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| Co-Investigator(Kenkyū-buntansha) |
KASAHARA Koji Kyoto University Yoshida College, Professor, 教養部, 教授 (70026748)
DATE Eturo Kyoto University Yoshida College, Assistant Professor, 教養部, 助教授 (00107062)
KONO Norio Kyoto University Yoshida College, Assistant Professor, 教養部, 助教授 (90028134)
ASANO Kiyoshi Kyoto University Yoshida College, Professor, 教養部, 教授 (90026774)
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| Project Period (FY) |
1987 – 1988
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| Keywords | Cauchy-Kowalewski theorem / Nash-Moser theorem / Boltzmann equation / Arithmetic-geometric mean / Central limit theorem / Log-log law / 高木関数 |
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| Research Abstract |
Our research consists of the unfied study of geometry and analysis, in connection with the geometric structure on a differentiable manifold. Below we state the summary of the results obtained. In the fluid dynamics,which play fundamental role in physics as well as in technology, we view the fluid as continuum, for the equation of the fluid flow as well as its solutions their asymptotic behavior were clarified[1]. When the initial data is sufficiently close to an equilibrium state, it is shown that the Boltzmann equation has a unique global solution[4]. An extention of the non-linear Cauchy-Lowalewski theorem is obtained with a simplified proof[2]. An simplified and elegant treatment of Nash-Moser implict function theorem was given[3]. An alternate proof of Cox-Geppert theorem on arithmetic-geometric mean was obtaind[5]. As for the investigations relating to the probability theory, for the multiplicative system of random variables, functional central limit theorem and log-log law were proved[6]. Generalized Takagi functions are investigated[7],a necessary and sufficient condition for an self-affine function is absolutely continuous with respect to lebesgues measure and surface filling functions, defined by two self-affine functions, are discussed[8]. In connection with the statistical mechnics, the local height probabilities of certain integrable SOS-modelzare obtained in terms of modular forms[9]. The local height probabilities of certain solvable SOS-model are calculated in terms of modular functions[10]. The startriangle relations and some combinatorial identities[11]. As stated above, not only the contents of our research cover a vast variety of mathematical science, but also they are excellent results on the topics of up-to-date. The research is continued now inceantly, and therefore, further developments can be expected.
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